Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations

TL;DR

This paper explores properties and characterizations of multidimensional compact almost automorphic functions, demonstrating their invariance under integral operators and applying these results to analyze solutions of Poisson and heat equations.

Contribution

It introduces two new characterizations of compact almost automorphic functions in multiple dimensions and studies their invariance under integral operators with automorphic kernels.

Findings

New characterizations for -multi-almost automorphic functions

Invariance of these functions under integral operators

Applications to Poisson and heat equations

Abstract

Let \(\mathcal{G}\) be a non-empty subset of the Euclidean space \(\mathbb{R}^m\) (\(m \geq 1\)). This work is dedicated to further exploring the properties of \(\mathcal{G}\)-multi-almost automorphic functions defined on \(\mathbb{R}^m\) with values in a Banach space \(\mathbb{X}\). Using the theory of \(\mathcal{G}\)-multi-almost automorphic functions, we provide two new characterizations of compact almost automorphic functions. In the first characterization, \(\mathcal{G}\) corresponds to the lattice subgroup \(\mathbb{Z}^m \subset \mathbb{R}^m\); in the second, \(\mathcal{G}\) is taken to be a dense subset of \(\mathbb{R}^m\). Furthermore, we establish the invariance of the space of bounded and compactly \(\mathcal{G}\)-multi-almost automorphic functions under integral operators with Bi-almost automorphic kernels. Finally, we present applications to the analysis of the almost automorphic dynamics of Poisson's equation and the heat equation.